90 A STUDY IN PUBLIC FINANCE „. „

the income-utility curve of the members of our community
(all of whom are assumed to be alike) is of this character or
of some other defined character by any process of general
reasoning. Nor is the Weber-Fechner law as tg physical
stimuli and the reactions found to be associated with them in
physiological laboratories of direct relevance to our problem,
though it affords a suggestive analogy. The only procedure
available is to ask ourselves directly a series of questions in
the form : Given that a £10 cut from £100 income involves
so much sacrifice to the representative man, what size of cut
from an £800, or a £1000, or a £10,000 income would involve
about the same amount of sacrifice ? The questions must be
put carefully. We are concerned, not with a single-event tax,
but with continuing tax systems. Therefore, when we speak
of " cuts " from incomes of different sizes, we must not imagine
the people affected to have developed their life and tastes to
fit with the incomes named. That would run counter to our
supposition that everybody is to be regarded as a representa-
tive man in respect of tastes. Tastes must be taken as given
and alike for all. An accurate formulation of our questions,
therefore, is : Given the difference in satisfactions yielded to
a representative man by a £200 and a £190 income respectively,
what is the income, the difference between which and £1000
income will represent that amount of satisfaction ? ; and so
on for all other incomes. It is extraordinarily difficult to give,
even within wide limits, any confident answer to this type of
question. I feel fairly certain that the gap between £200 and
£190 means more than the gap between £1000 and, say, £980.
But does it mean more or less than the gap between £1000
and £970, or between £1000 and £940 ? I hesitate to say.
Bernoulli's familiar hypothesis amounts, in effect, to the
proposition that, in respect of incomes in excess of what is
required to yield the necessaries of life, the income-utility
curve of the representative man is a rectangular hyperbola.
Prima facie, this seems not implausible — much more plausible
than Cramer's hypothesis, which Marshall mentions, that the

and it being taken for granted that the curve in debate slopes downwards
towards the right, let h be any constant fraction. Then, in order that

AC rjac

I F(a) - I F(Aa?) may be constant for all values of x, it is necessary that

) ; a condition which, in general, implies that xF(x) is constant
for all values of x.